Exact Coherent Structures and Data-Driven Modeling in a Two-Dimensional Fluid

Turbulence is one of the most ubiquitous features of the world around us. Its signatures can be found at every scale, in the small eddies of streams to the vortex wakes of airliners, from ocean currents to the cosmic swirls of interstellar gases. Fluids like these have been studied for centuries, but while many advances have been made toward understanding the captivating patterns that they create, predicting the evolution of turbulent fluids remains one of the most notoriously difficult unsolved problems in classical physics. The Navier-Stokes equation is a deterministic, high-dimensional partial differential equation which allows us to make limited predictions about fluids under particular conditions. However, a characteristic feature of turbulence is that it is chaotic, meaning the evolution of a turbulent fluid is highly sensitive to its initial conditions. In practice these initial conditions can never be measured precisely enough to make long-term predictions tractable, and often measurements cannot be made of every physical quantity that goes into the Navier-Stokes equation. Physicists have thus turned recently to developing data-driven computational fluid models to gather statistics about recurring patterns that guide the flow's evolution, which can then be used to characterize an experimental flow. In this dissertation, two data-driven approaches are explored in the context of a shallow, driven fluid flow, an experimental approximation of two-dimensional (2D) Kolmogorov flow. The first approach provides experimental evidence for the statistical role of exact, unstable solutions of the Navier-Stokes equation known as exact coherent structures. In particular, periodic orbits are shown to play an important role in guiding the dynamics of a turbulent flow, as the fluid flow spends a large amount of time shadowing the most relevant orbits as predicted by periodic orbit theory. In the second approach, a weak formulation of the symbolic regression algorithm is used to develop a model of the 3D fluid using only a 2D approximation of the velocity field. The model can then be used to recreate the pressure and forcing fields, yielding a modified, quasi-2D Navier-Stokes equation that governs the flow and agrees with the first-principles model derived in previous studies. Finally, as fluid properties change, the variation in the coefficients of this quasi-2D model are also in agreement with predictions from previous work and provide a useful diagnostic tool for common experimental errors. The substantial progress provided by this dissertation suggests that physics-informed data-driven analysis of turbulent flows provides an important validation of existing models and theories.Ph.D

Capturing Turbulent Dynamics and Statistics in Experiments with Unstable Periodic Orbits

In laboratory studies and numerical simulations, we observe clear signatures of unstable time-periodic solutions in a moderately turbulent quasi-two-dimensional flow. We validate the dynamical relevance of such solutions by demonstrating that turbulent flows in both experiment and numerics transiently display time-periodic dynamics when they shadow unstable periodic orbits (UPOs). We show that UPOs we computed are also statistically significant, with turbulent flows spending a sizable fraction of the total time near these solutions. As a result, the average rates of energy input and dissipation for the turbulent flow and frequently visited UPOs differ only by a few percent.Comment: Accepted for publication in Phys. Rev. Let